Simply rifle through Anuriel's dresser to grab the incriminating letter without attracting too much attention to yourself to get it taken away by the nosy guards and show the letter to Anuriel. She immediately asks to retreat to the privacy of her room, where the two of you arrive at an agreeable compromise.

The compelling evidence in question is Raerek's Amulet of Talos, which can be found in the dresser near the bed in his room.

The room is guarded by a single patrolling city guard and sneakery or trickery, such as a clever use of the Whirlwind Sprint will be required to get in. The room itself can be found just past the Jarl's blacksmith.

After reporting to your respective higher-ups, you'll need to meet with either Hadvar (Imperial) or Ralof (Stormcloaks). Follow them and dispose of the enemies to finish this quest.